Problem: Simplify; express your answer in exponential form. Assume $t\neq 0, x\neq 0$. $\dfrac{{t^{-3}}}{{(t^{3}x^{-1})^{-3}}}$
Answer: To start, try working on the numerator and the denominator independently. In the numerator, we have ${t^{-3}}$ to the exponent ${1}$ . Now ${-3 \times 1 = -3}$ , so ${t^{-3} = t^{-3}}$ In the denominator, we can use the distributive property of exponents. ${(t^{3}x^{-1})^{-3} = (t^{3})^{-3}(x^{-1})^{-3}}$ Simplify using the same method from the numerator and put the entire equation together. $\dfrac{{t^{-3}}}{{(t^{3}x^{-1})^{-3}}} = \dfrac{{t^{-3}}}{{t^{-9}x^{3}}}$ Break up the equation by variable and simplify. $\dfrac{{t^{-3}}}{{t^{-9}x^{3}}} = \dfrac{{t^{-3}}}{{t^{-9}}} \cdot \dfrac{{1}}{{x^{3}}} = t^{{-3} - {(-9)}} \cdot x^{- {3}} = t^{6}x^{-3}$.